Analysis of Vibration Characteristics for a Rotor-Bearing System Using Distributed Spring and Damper Model

In this paper, we modelled the journal beatings as distributed springs and dampers to investigate the influence of beating width on the vibration characteristics of a rotor-beating system. A quadratic function is used as a shape function for hydrodynamic journal bearings. And a trapezoidally or a quadratically distributed model is used as a shape function for hydrostatic journal bearings. A finite element method is applied for analyzing the vibration characteristics of a rotor-bearing system.


INTRODUCTION
present, the development of turbomachinery tends to increase the rotating speed for the purpose of decreasing the weight to power ratio of machines.So, the vibration is one of the most important problems to be solved.In analyzing the dynamic behavior of rotor- bearing systems, various mathematical models have been developed and used over the past few decades.A finite element approach is one of the most widely used methods for this purpose.
In the early finite element investigations, many effec- tive models considering gyroscopic effect, axial force, torque, internal viscous damping, internal hysteretic damping and shear effects for uniform or tapered shafts were developed by Nelson [1980], 0zguven and Ozkan [1984], Rouch and Kao [1979], and Gmtir and Rodrigues [1991].The bearings were simplified as point supported or uniformly distributed springs and dampers by Mourelatos and Parsons [1987].It is a reasonable assumption when using ball bearings or short width journal beatings.
However, in the case of a long width journal bearing the pressure created in journal is different along the axial direction as well as radial direction.So, when the bearing width increases for the sake of large bearing load capacity, the bearing force is no more a point force.The pressure distribution in axial direction of a hydrodynamic journal bearing is quadratic function or trapezoidal function for a hydrostatic journal bearing.
In the present study, journal bearings are modelled as quadratically and trapezoidally distributed springs and dampers along the shaft.For these models natural fre- quencies for various design parameters are calculated and compared to each other.And the stability limits of the rotating speed are calculated for an example rotor using two types of support model.

MODELING AND GOVERNING EQUATIONS
The finite element equation for a typical rotor-bearing system can be written as From these approximated shape functions we can rewrite equation (2) as where subscript, B, denotes the bearing element.
The stiffness and damping matrices of a bearing, shown in FIGURE 1, can be written as [ ] Kyyfi(s) Kyzfi(s) [K] .J [c-IT Kzyfi(S)  Kzzfi(s) [K] J []T[ kyy(S) k y z ( S ) ] kzy(S kzz (S)   [ ] (4) [C] 3[]T[ cyy(S) C y z ( S ) ] Czy(S Czz (S)   []ds (2) where s denotes an axial distance along element and [y] the matrix of the translational displacement shape func- tion of a finite shaft element defined by equation (3), {V W} T [xI/'] {q, q2 q3 q4 q5 q6 q7 q8} T (3) Shape functions, kij and cij, were assumed as Dirac delta function(assumed a bearing as point supported one) or constant value(assumed as uniformly distributed one).However, a typical bearing analysis shows that the distribution shapes of springs and dampers are different from those assumptions.FIGURE 2 and FIGURE 3 show the shapes of dynamic coefficients of typical hydrodynamic and two- line feed hydrostatic journal bearings, respectively.We approximate these shape functions as quadratic o,r trap- ezoidal functions, which are shown in FIGURE 4. These approximated shapes are shown by dashed lines in FIGURE 2 and FIGURE 3.
For a two-line feed hydrodynamic bearing we can rewrite equation (2) as follows.

NONDIMENSIONALIZATION
In this study, we consider the uniform shaft supported by two journal bearings as in FIGURE 5 to estimate the effects of bearing width on vibration characteristics.The governing equations of this system are given by equation (1).
The equations of this model shown in FIGURE 5 can be rewritten by using the following dimensionless quan- Then, using the parameters in equation ( 6) the nondi- mensionalized equations are derived as The generalized eigenvalue problem can be caed out, after reaanging equation (8) into a system of first order  where (10) The damped natural frequencies of the system are then obtained by finding the eigenvalues of the dynamic matrix [D] which is given by [D] [A] - [B]   (11)

NUMERICAL EXAMPLES AND RESULTS
Example 1 In order to investigate the various effects of the rotor- bearing system on the natural frequencies, we use four different models.They are models supported by point, by uniform pressure, quadratically and trapezoidally distrib- uted pressure over the length of the bearing, respectively.From the earlier analyses of bearings, we can approxi- mate the magnetic bearings by uniformly distributed model, the hydrodynamic journal bearings by quadrati- cally distributed model and the two-line feed hydrostatic journal bearings by hybrid model, that is, uncoupled springs are approximated by trapezoidally .distributedand coupled springs and all dampers are approximated by quadratically distributed model.We use 9 shaft elements in point supported model and 8 elements in other models.All damping coefficients and the coupled terms of spring coefficients are set to zero and the uncoupled terms of spring coefficients are set by Cyy Czz Cy Czy Ky Kzy 0 (12) Kyy Kzz K B (13) Other data for these cases are shown in TABLE 1.We calculate the first two natural frequencies of the system varying bearing length ratio(B/L) for two differ- ent values of B/L.These results are shown in FIGUREs 6 and 7.Because the deflection shape of the shaft between the nodes is approximated by cubic function in finite element modeling, the stiffness of the distributed models is less than the point supported model.The uniformly distributed model is stiffer against rotation than other distributed models.So the natural frequencies of point supported and uniformly distributed models are overestimated from these results.Point supported and uniformly distributed models give similar results, and quadratically or trapezoidally distributed model gives similar results.(Some results are plotted as if they have same values.)So we can use alternative model for the vibration analysis.FIGURE 7 shows the difference ratio between point supported model and other models defined by to 100 1, 2, 3 (14 where 60kp is the k-th natural frequency of the point supported model and 6Oki is the k-th natural frequency of other models.From these results it can be observed that the difference is considerable as the bearing length ratio is increased.
Similar calculations are carried out with varying the stiffness ratio of bearings for the beating length ratio of 0.1.From FIGUREs 8 and 9 it can be seen that differ- ences between other models are increased as the bearing stiffness ratio is increased, but the difference ratio is almost independent of the beating stiffness ratio except for the 1st natural frequency of uniformly distributed model.

Example 2
We consider the rotor model which has three discs and supported by two cylindrical journal bearings as shown in FIGURE 10.The stability limit of the rotating speed is calculated for two types of support models, i.e., point supported model and quadratically distributed one.The results are shown in FIGURE 11.(The results of Glinicke et al. [1980] are used as the rotor-dynamic coefficients of the cylindrical journal bearings for some B/D ratios.The data of the discs are shown in TABLE 2.) From these results it is found that the stability limit of distributed model is lower than that of point supported one in the case of small B/D.The reason is that the logarithmic decrements of distributed model are smaller than those of point supported one.1, (a) Case FIGURE 8 First two natural frequencies for some bearing stiffness ratio..:..:::.:.'.

CONCLUSIONS
The finite element approach is used in analyzing the influence of bearing width on the vibration characteris- tics of rotor-bearing systems where various models of bearing supports are used.It is suggested that hydrody- namic and hydrostatic journal bearings should be approximated by quadratically and trapezoidally distrib- uted springs and dampers.Then, the first two natural frequencies of the uniform shaft, supported by two bearings, are calculated and compared for the point supported model, the uniformly distributed model and hydrostatically or hydrodynamically distributed model.And we calculate the stability limit of the simple rotor model supported by two cylindrical journal bearings.
From this analysis, it is observed that the point supported or uniformly distributed bearing models overestimate the natural frequencies and the logarithmic decrements.So we should consider the effects of bearing width as the beating length ratio, B/L, is increased.However, in the present studies the bearing data are calculated on the assumption that the portion of the shaft in the bearing is rigid, so iterative method between flexible rotor analysis and bearing analysis will give more realistic results for a real rotor bearing system.(It is in progress.)

Acknowlegements
This research was carried out at the Turbo and Power Machinery Research Center and was funded by the Korea Science and Technology Engineering Foundation.Their supports are gracefully acknowledged.

FIGURE
FIGUREFinite shaft element in a bearing,

FIGURE 3 FIGURE 4
FIGURE 3 Distribution shapes of springs and dampers for two-line feed hydrostatic bearing.
FIGURE 9Difference ratio for some bearing stiffness ratio.
shape function of bearing stiffness and damping to the direction i, D.C. HAN ET AL.